Cambridge Journal of Mathematics

Volume 1 (2013)

Number 2

Moduli of $p$-divisible groups

Pages: 145 – 237



Peter Scholze (Mathematisches Institut der Universität Bonn, Germany)

Jared Weinstein (Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, U.S.A.)


We prove several results about moduli spaces of $p$-divisible groups such as Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of $p$-adic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of $p$-divisible groups over $\mathcal{O}_C$, where $C$ is an algebraically closed complete extension of $\mathbb{Q}_p$, in the spirit of Riemann’s classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonné module functor for $p$-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).

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